# Lecture 22 11/28/2007 Meenakshi Narain - Physics 5

Lecture 22 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 1 Announcements Homework due TODAY at the end of lecture Homework 12 based on ch 12 & 13, due on Dec 5th. I may post some practice problem suggestions for ch 14 and 15. Final Exam on Thursday, Dec 13, 9am-12pm, in BARHOL 168 Close book, close lecture notes Chapters 1-15, with emphasis on post midterm 2 material Emphasis of the exam: problem solving (6-8 problems)

Simple calculator and one-page formula sheet allowed Questions? Suggestions? 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 2 Simple Harmonic Motion Chapter 15 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 3 A block of mass 1 kg is attached to a spring with k=100 N/m and free to move on a frictionless horizontal surface. At t=0, the spring is extended 5 cm beyond its equilibrium position and the block is moving to the left

with a speed of 1 m/s. What is the displacement of the block as a function of time? 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 4 x(t ) A cos(t ) What are the values of A, , and ? A, are determined by initial conditions (x,v at t=0) they are not properties of the oscillating system. x t 0 A cos v t 0 A sin v 0 x 0 tan A x(t ) A cos(t ) x 0 cos and

k m k 10 / s m v(0) -1 m/s tan 2 x(0) 10/s 0.05 m 1.1 rad x(0) 0.05 m A 0.11 m cos 0.45 x(t) 0.11 m cos(10 /s t 1.1) 11/28/2007

Meenakshi Narain - Physics 5 - Lectu re 22 5 Three 10,000 kg ore cars are held at rest on a 30 o incline using a cable. The cable stretches 15 cm just before a coupling breaks detaching one of the cars. Find (a) the frequency of the resulting oscillation of the remaining two cars and (b) the amplitude of the oscillation. 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 6 m = 10000 kg d = 0.15 m first compute spring constant of rope: k

3mg sin N 9.8 105 d m then compute frequency of oscillation: f 1 2 2 k 1 2m 2 3g sin 1.1 Hz 2d the amplitude equals the original displacement from the equilibrium position of

two cars on the rope: A = d d0 = 0.05 m 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 7 A 10 g bullet strikes a 1 kg pendulum bob that is suspended from a 10 m long string. After the collision the two objects stick together and the pendulum swings with an amplitude of 10o. What was the speed of the bullet when it hit the bob? 10o 10 g 1.0 kg 1.0 kg

v=? 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 8 Energy of simple pendulum 2 U mgh mg L L cos mgL using 2 max cos 0 t 1 2 1 2 E I mgL with max 0 sin 0 t 2 2 small x2 x4 cos x 1

... 2! 4! g 0 2 L 1 1 1 2 2 2 K I 2 mL2 max 0 sin 2 0t mgL max sin 2 0t 2 2 2 1 1 2 U mgL 2 mgL max cos 2 t 2

2 1 2 E K U mgL max 2 1 1 m 2 2 E MgL max 1 kg 9.8 2 10 m 0.17 1.5 J 2 2 s Get velocity of bob at equilibrium point from total energy 2E vf 1.7 m/s p f Mv f 1.7 kgm/s M get velocity of bullet from momentum conservation p f 1.7 kgm/s vi

170 m/s m 0.01 kg 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 9 CO2 molecule Carbon dioxide is a linear molecule. The C-O bonds act like springs. This molecule can vibrate such that the oxygen atoms move symmetrically in and out, while the carbon atom is at rest. The frequency of this vibration is observed to be 2.83x1013 Hz. What is the spring constant of the C-O bond? In which other way can the molecule vibrate and at what frequency? If the amplitude is the same, in which mode is the energy of the molecule higher? m=12u

11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 m=16u 10 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 11 CO2 molecule a) 1 f 2 2

k k 4 2 f 2 mO mO 2 4 2.83 10 Hz 16u 1.67 10 27 kg/u 844 N/m 2 b) F 2kx k ' 2k f ' c) 13 ' 1 2 2 k' 1 mC 2

2k mC 1 2 844 N/m 13 4.62 10 Hz 27 2 12u 1.67 10 kg/u Parallel springs See Lecture 10 2 E kA 1 1

E kA 2 1 2 2 E k ' A kA 2 2 2 the energy is the same for both modes. 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 12

Which of the following spring arrangements will oscillate with the smallest angular frequency? Assume that all springs are identical. (1) (2) (3) (4) 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 13 Driven Harmonic Oscillator What happens if you have an oscillator, such as a mass on a spring, where an external force is acting on the system? Example: Motion of a building or bridge during an earthquake Essentially all objects have one or more natural

frequencies that they will oscillate at if they are initially displaced from equilibrium Example: mass on a spring has a natural frequency given by k 0 m If an oscillating external force is applied with angular frequency close to the natural frequency 0, the results can be dramatic 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 14 Driven Harmonic Oscillator To get motion, use Newtons 2nd law: ma Fnet bv kx Fext (t )

d 2x dx m 2 b kx Fext (t ) dt dt Fext (t ) d 2x dx 2 2 x 0 dt 2 dt m b 2m

k 0 m Suppose the external force is sinusoidal: Fext (t ) F0 cos( t ) F0 d 2x dx 2 2 0 x cos( t ) 2 Eventually, the objects motion will oscillate with frequency dt dt m

w since thats the frequency of the applied force 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 15 Driven Harmonic Oscillator Solution for driven harmonic oscillator is somewhat more complicated than what we have done so far in Physics 5 With some one can solve the equation of motion d 2 xeffort, dx F 2 0 2 0 x cos( t ) 2 dt

dt m x x0 cos( t ) xDamped (t ) where the frequency F0 1 and amplitude are given by: x0 m (02 2 ) 2 4 2 2 2 tan( ) 2 02 xDamped (t ) x0 e t cos( ' t 0 ) 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 ' 02 2

16 Driven Harmonic Oscillator Example: Mass on spring Notice how amplitude and phase change as the frequency of the external force crosses the natural frequency F0 x0 m 1 (02 2 ) 4 2 2 2 tan( ) 2 0 2 Example: Vibrations of a solid object Solid objects typically have one or more natural frequencies that they oscillate at If the damping is small, large oscillations occur when driven at these natural frequencies These vibrations can be a considerable source of stress on the

object! 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 17 Oscillations Summary We get periodic motion when force acts to push object back towards equilibrium position Many problems exhibit simple harmonic motion d 2x 2 x(t ) xm cos( t )

x 2 dt Energy exchanged between kinetic and potential energy, total mechanical energy unchanged for undamped oscillations Correspondence between simple harmonic motion and uniform circular motion Amplitude of oscillations decays with a damping force Driven oscillations exhibit a resonance at the natural frequency 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 18 Fluids Chapter 14 11/28/2007 Meenakshi Narain - Physics 5 - Lectu

re 22 19 Fluids What is a fluid? A substance that flows Examples include liquid, gas, plasma, etc A simple fluid can withstand pressure but not shear Density m V Density Unit: kg/m3 Examples: density of water 1000 kg/m3 ; air 1.21 kg/m3 A materials specific gravity is the ratio of the density of the material to the density of water at 4C. What is special about water at 4C?

Water is most dense at that temperature. Aluminum has a specific gravity of 2.7 it is 2.7 times more denser than water at 4C. 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 20 A table of densities Material Density (kg/m3) Interstellar space 10-20 Air (20C, 1 atm.) 11/28/2007 1.21

Water (4C, 1 atm.) 1000 Sun (average) 1400 Earth (the planet) 5500 Iron 8700 Mercury (the metal) 13600 Black hole 1019 Meenakshi Narain - Physics 5 - Lectu

re 22 21 Pressure Pressure p F A Unit: pascal (Pa)=N/m2 Examples: 1 atm=1.01x105 Pa=760 torr=14.7 lb/in2 Torricelli (torr) is defined as the pressure of mm Hg. Blood pressure: 70/120 torr At any point in a fluid at rest, the pressure is the same in all directions. If this were not true there would be a net force on the fluid and it could not be at rest. The force due to fluid pressure acts perpendicular to

any surface. Else there would be a force component along the surface which would accelerate the fluid. 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 22 Atmospheric pressure At atmospheric pressure, every square meter has a force of 100,000 N exerted on it, coming from air molecules bouncing off it! Why dont we, and other things, collapse because of this pressure? We have an internal pressure of 1 atmosphere. Objects like tables do not collapse because forces on top surfaces are balanced by forces on bottom surfaces, etc. 11/28/2007

Meenakshi Narain - Physics 5 - Lectu re 22 23 Fluid-Statics Static equilibrium F2 F1 mg p2 A p1 A Vg p1 A A( y1 y2 ) g p2 p1 ( y1 y2 ) g p1 gy1 p2 gy2 const A simpler expression p p0 gh Where p0 is the pressure at the surface, and h is depth of the liquid The pressure at any point in a fluid is determined by the density of the fluid and the depth. It does not depend on any horizontal dimension of the fluid or its container. It also does not depend on the shape of the container. 11/28/2007

Meenakshi Narain - Physics 5 - Lectu re 22 24 Measuring pressure The relationship between pressure and depth is exploited in manometers (or barometers) that measure pressure. P2 P1 gh A standard barometer is a tube with one end sealed. The sealed end is close to zero pressure, while the other end is open to the atmosphere. The pressure difference between the two ends of the tube can maintain a column of fluid in the tube, with the height of the column being proportional to the pressure difference. pressure at bottom of column = atmospheric pressure

11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 25 Mercury/Water barometer Mercury: atmospheric pressure pushes Hg column up unit mm-Hg (=torr) 1 mm Hg gh 13.6 10 3 kg/m 3 9.8 m/s 2 0.001 m 133 Pa Thus Atmospheric pressure pushes the Hg column up by 101.3 kPa/133 Pa/mm = 760 mm Water: 1 mm H 2 O gh 10 3 kg/m 3 9.8 m/s 2 0.001 m 9.8 Pa Thus atmospheric pressure pushes the water column up by 101.3 kPa/9.8 Pa/mm = 10.3 m

another unit: 1 bar = 105 N/m2 in calculations only use N/m2 = Pa (SI unit) 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 26 Gauge Pressure Gauges measure pressure relative to atmospheric pressure absolute pressure = gauge pressure + atmospheric pressure manometer (height of column of liquid measures gauge pressure) 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 27 Blood pressure

A typical reading for blood pressure is 120 over 80. What do the two numbers represent? What units are they in? 120 mm Hg (millimeters of mercury) is a typical systolic pressure, the pressure when the heart contracts. 80 mm Hg is a typical diastolic pressure, the blood pressure when the heart relaxes after a contraction. 760 mm Hg is typical atmospheric pressure. The blood pressure readings represent gauge pressure, not absolute pressure they tell us how much above atmospheric pressure the blood pressure is. 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 28 A delicious drink sits on the patio. From your balcony several stories up you

manage to lower a straw into the glass, which is 15 m below you. Can you syphon up the drink? 15 m yes, but I will have to suck really hard (2) probably not, but a vacuum pump could (3) no, this is not possible (4) I dont know (1) 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 29 Water pressure At the surface of a body of water, the pressure you experience is atmospheric pressure. Estimate how deep

you have to dive to experience a pressure of 2 atmospheres. P2 P1 gh 200000 Pa 100000 Pa (1000 kg/m3 ) (10 N/kg) h h works out to 10 m. Every 10 m down in water increases the pressure by 1 atmosphere. 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 30 Rank by pressure A container, closed on the right side but open to the atmosphere on the left, is almost completely filled with water, as shown. Three points are marked in the container. Rank these according to the pressure at the points, from highest pressure to lowest.

A=B>C B>A>C B>A=C C>B>A C>A=B some other order 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 31 Blaise Pascal (1623-1662) A change in the pressure

applied to an enclosed incompressible fluid is transmitted undiminished everywhere in the fluid and to the walls of the container p pext gh p pext 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 32 A container is filled with oil and fitted on both ends with pistons. The area of the left piston is 10 mm2; that of the right piston is 10,000 mm2. What force must be exerted on the left piston to keep the 10,000 N car on the right at the same height? =10000 N (1)

(2) (3) (4) (5) 10 N 100 N 10,000 N 106 N 108 N =? =10 mm2 =10000 mm2 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 33 Pascals Principle p pext gh

p pext Hydraulic lever (see diagram on right) Something has to give Since the liquid is incompressible, the volume drop on the left is equal to that rises on the right, ie. Ai hi Ao ho 11/28/2007 therefore ho A0 hi Ai Meenakshi Narain - Physics 5 - Lectu re 22 Fi Fo

p Ai Ao Ao Fo Fi Ai 34 Pascal placed a long thin tube vertically into a wine barrel. When the barrel and tube were filled with water to a height of 12 m, the barrel burst. (a) what is the mass of the water in the tube? (b) what is the net force exerted onto the lid of the barrel? 11/28/2007

Meenakshi Narain - Physics 5 - Lectu re 22 35 Imagine holding two identical bricks under water. Brick A is just beneath the surface. Brick B is at a greater depth. The force needed to hold brick B in place is (1) larger (2) the same as (3) smaller than the force required to hold brick A in place 11/28/2007

Meenakshi Narain - Physics 5 - Lectu re 22 36 The Buoyant Force With fluids, we bring in a new force. The buoyant force is generally an upward force exerted by a fluid on an object that is either fully or partly immersed in that fluid. Lets survey your initial ideas about the buoyant force. 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 37 The Buoyant Force A wooden block with a weight of 100 N floats exactly 50% submerged in a particular fluid. The upward buoyant force exerted on the block by the fluid

has a magnitude of 100 N has a magnitude of 50 N depends on the density of the fluid depends on the density of the block depends on both the density of the fluid and the density of the block 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 38 Learning by Analogy Our 100 N block is at rest on a flat table. What is the normal force exerted on the block by the table?

To answer this, we apply Newtons Second Law. There is no acceleration, so the forces balance. 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 39 Apply this to Buoyant force Apply the same method when the block floats in the fluid. What is the magnitude of the buoyant force acting on the block? To answer this, we apply Newtons Second Law. There is no acceleration, so the forces balance. 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22

40 Reviewing the normal force We stack a 50-newton weight on top of the 100 N block. What is the normal force exerted on the block by the table? To answer this, we apply Newtons Second Law. There is no acceleration, so the forces balance. The block presses down farther into the table (this is hard to see). 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 41 Buoyant force We stack a 50-newton weight on top of the 100 N block. What is the buoyant force exerted on the block by the fluid?

To answer this, we apply Newtons Second Law. There is no acceleration, so the forces balance. The block presses down farther into the fluid (this is easy to see). 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 42 Apply Newtons Second Law Even though we are dealing with a new topic, fluids, we can still apply Newtons second law to find the buoyant force. 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 43 Three Beakers

The wooden block, with a weight of 100 N, floats in all three of the following cases, but a different percentage of the block is submerged in each case. In which case does the block experience the largest buoyant force? 4. The buoyant force is equal in all three cases. 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 44 Three Beakers What does the free-body diagram of the block look like? What is the difference between these fluids? The density 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22

45 A block of weight mg = 45.0 N has part of its volume submerged in a beaker of water. The block is partially supported by a string of fixed length. When 80.0% of the blocks volume is submerged, the tension in the string is 5.00 N. What is the magnitude of the buoyant force acting on the block? 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 46 Apply Newtons Second Law The block is in equilibrium all the forces balance. Taking up to be positive: V Fy may

FB FT mg 0 FB mg FT 45.0 N 5.00 N 40.0 N 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 47 Water is steadily removed from the beaker, causing the block to become less submerged. The string breaks when its tension exceeds 35.0 N. What percent of the blocks volume is submerged at the moment the string breaks? 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 48 Apply Newtons Second Law

The block is in equilibrium all the forces balance. Taking up to be positive: Fy may FB FT mg 0 FB mg FT 45.0 N 35.0 N 10.0 N The buoyant force is proportional to the volume of fluid displaced by the block. If the buoyant force is 40 N when 80% of the block is submerged, when the buoyant force is 10 N we must have 20% of the block submerged. 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 49 After the string breaks and the block comes to a new

equilibrium position in the beaker, what percent of the blocks volume is submerged? what does the free-body diagram look like now? 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 50 Apply Newtons Second Law The block is in equilibrium all the forces balance. Taking up to be positive: Fy may FB mg 0 FB mg 45.0 N The buoyant force is proportional to the volume of fluid displaced

by the block. If the buoyant force is 40 N when 80% of the block is submerged, when the buoyant force is 45 N we must have 90% of the block submerged. 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 51 Archimedes Principle While it is true that the buoyant force acting on an object is proportional to the volume of fluid displaced by that object. But, we can say more than that. The buoyant force acting on an object is equal to the weight of fluid displaced by that object. This is Archimedes Principle. FB mdispg fluid Vdisp g m mass is our symbol for mass density:

V volume 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 52 A Floating Object When an object floats in a fluid, the downward force of gravity acting on the object is balanced by the upward buoyant force. mg fluid Vdisp g object Vobject g fluid Vdisp g object Vobject fluid Vdisp object Vdisp fluid Vobject Looking at the fraction of the object submerged in the fluid tells us how the density of the object compares to

that of the fluid. 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 53 Beaker on a Balance A beaker of water sits on a scale. If you dip your little finger into the water, what happens to the scale reading? Assume that no water spills from the beaker in this process. 1. The scale reading goes up 2. The scale reading goes down 3. The scale reading stays the same 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 54

Three Blocks We have three cubes of identical volume but different density. We also have a container of fluid. The density of Cube A is less than the density of the fluid; the density of Cube B is exactly equal to the density of the fluid; and the density of Cube C is greater than the density of the fluid. When these objects are all completely submerged in the fluid, as shown, which cube displaces the largest volume of fluid? 1. 2. 3. 4. Cube A Cube B Cube C The cubes all displace equal volumes of fluid 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 55

Three Blocks Each cube displaces a volume of fluid equal to its own volume, and the cube volumes are equal so the volumes of fluid displaced are all equal. 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 56 Three Blocks Which object has the largest buoyant force acting on it? 1. 2. 3. 4. Cube A Cube B

Cube C The cubes have equal buoyant forces 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 57 Three Blocks Each cube displaces an equal volume of the same fluid, so the buoyant force is the same on each. 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 58 Two identical glasses are filled to the brim with water. One of

the two glasses has a ball floating in it. Which glass weighs more? 1. The glass without the ball 2. The glass with the ball 3. The two weigh the same 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 59 A boat carrying a large boulder is floating in a lake. The boulder is thrown overboard and sinks. What happens to the water level in the lake (relative to the shore)? (1) it sinks (2) it rises (3) it remains the same 11/28/2007

Meenakshi Narain - Physics 5 - Lectu re 22 60 Cartesian diver The diver is an object in a sealed container of water. Air in the diver makes it buoyant enough to barely float at the water's surface. When the container is squeezed, the pressure compresses the air and reduces its volume. This permits more water to enter the diver, resulting in it being less buoyant and sinking. regular coke and diet coke 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 61 The origin of the buoyant force The net upward buoyant force is the vector sum of the

various forces from the fluid pressure. P2 P1 gh Because the fluid pressure increases with depth, the upward force on the bottom surface is larger than the downward force on the upper surface of the immersed object. Fnet P A fluid gh A fluid gV This is for a fully immersed object. For a floating object, h is the height below the water level, so we get: 11/28/2007 Fnet fluid gVdisp Meenakshi Narain - Physics 5 - Lectu re 22 62

When the object goes deeper If we displace the object deeper into the fluid, what happens to the buoyant force acting on it? Assume the fluid density is the same at all depths. The buoyant force: 1. 2. 3. increases decreases stays the same 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 63 When the object goes deeper P2 P1 gh

If the fluid density does not change with depth, all the forces increase by the same amount, leaving the buoyant force unchanged! 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 64 Archimedes Principle Buoyant force FB m f g Objects that float Dry wood, ice, some plastics, oil, wax (candles) Boats made of woods, ceramic, steel, or any other materials, as long as they are hollow enough Objects that sink

Rocks, sands, clay, metal, etc. Any material with density larger than water Apparent weight (example: submerged object weighs less Apparent weight=actual weight - buoyant force What is your apparent weight in water? (no more than a few pounds!) 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 65 Unbalancing the forces If we remove the balance between forces, we can produce some interesting effects. Demonstrations of this include: 1. 2. The Magdeburg hemispheres (see below) Crushing a can

11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 66 Crush a can Remember that this is just the collective effect of a bunch of air molecules! 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 67 Summary Density and pressure of fluids p p0 gh

Air pressure, blood pressure and underwater pressure Pascals Principle p pext gh p pext Archimedes Principle B f F m g 11/28/2007 Meenakshi Narain - Physics 5 - Lectu re 22 68

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